Microwave absorption and resistively shunted Josephson junctions in high temperature CuO superconductors
J.S. Ramachandran, M.X. Huang and S.M. Bhagat Department of Physics, Center for Superconductivity Research, University of Maryland, College Park, MD 20742, USA
K. Kish and S. Tyagi Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 11004, USA
Received 4 August 1992
We report that the field dependence of the magnetoabsorption (virgin curve) in all pristine granular CuO type (HTSC) super- conductors follows a one-parameter expression. This "universal" result is combined with previous measurements on field and temperature dependences of microwave absorption in HTSC to demonstrate that a simple model based on resistively shunted Josephson junctions is adequate to qualitatively account for almost all the observations on powders, pellets, thin films and single crystals.
1. Introduction
It is generally recognized that a careful study of the microwave losses yields fundamental information about a superconductor [ 1 ]. This laboratory has been involved in systematic investigations of the micro- wave absorption, both as function of temperature and magnetic field, in a variety of samples of the CuO superconductors. Several previous results [ 2-11 ] are worth recalling in order to motivate the present pa- per wherein we plan to demonstrate that weak links, treated as resistively shunted Josephson junctions (RSJ), play an essential role in explaining a wide range of the observations.
First consider c-axis oriented YBCO thin films [11]. ( 1 ) T mw, which marks the sharp drop in microwave absorption (fig. 4(a) below) is lower than T~ ~, ob- tained by magnetic measurements, and the discrep- ancy increases as the microwave frequency is increased. (2) The field induced magnetoabsorption is signif- icantly smaller than that in powders. It becomes size- able only when B> 50 mT and (T/T¢) > 0.9. Its field and temperature dependence can be well understood [ I 1 ] by using conventional fluxon dynamics with
parameter values well in accord with other data. (3) The zero-kelvin London length 2~ w, is larger than the accepted values obtained by other techniques.
Second, we record the results obtained with mi- cron-sized powders. ( l ) The sharpness of the transition [ 3 ] is, by and large, controlled by the ratio u = R / 2 (where R is the average radius), the T variation of 2 being set by the two-fluid expression, that is,
2=20[ 1 -- (T/Tc) 4 ] - 1 / 2 (1 )
However, (ref. [3] and table 2 below) the 20 values derived from the microwave data are significantly larger than those obtained from the T variation of the initial susceptibility of the same powders [ 8 ]. (2) The To, which marks the sharp drop in micro- wave absorption, (cf. fig. 1, ref. [2] ) is a few kelvin lower than Tc obtained from DC, or low frequency AC, magnetization studies [ 2,10 ]. (3) Whereas the magnetic isotherms are close to being linear (Meissner regime) the (magnetic) field induced absorption, is sizeable and the virgin curve [ 12] follows the empirical form,
152 J.S. Ramachandran et al. / Microwave absorption
(4) When the applied field is cycled, one invariably observes a "hysteresis" loop (of the type shown in fig. 3 below) whose characteristics are strongly in- fluenced by temperature [ 13 ] and the maximum field to which the specimen has been exposed previously.
Finally, we come to the new investigations. In ref. [ 12 ] it was noted, in passing, that some of the data on sintered samples could also be described ade- quately by eq. (2). It was therefore decided that a systematic study employing (a) powders of various sizes and prepared by different methods, (b) ag- glomerates and (c) sintered pellets, would be fruitful to establish the general applicability of eq. (2) to all grainy materials. The results of this investigation are reported below, and as we shall see, eq. (2) is es- sentially a universal virgin curve, no matter how the CuO superconductor is prepared and no matter that the zero-field-cooled magnetic isotherm deviates from linearity. The problem, therefore, reduces to obtaining a credible microscopic picture to account for the field scaling parameter H0 as well as its de- pendence on frequency and temperature.
Also, in order to gain further insight into the be- havior of less grainy specimens, the thin film inves- tigations have been expanded to include PrBCO- YBCO-PrBCO "sandwich" type structures. In ad- dition, we have carried out measurements on single crystals of ErBCO and YBCO. In the latter, the tran- sition widths are indeed narrow. However, T~W< Tmag and again the discrepancy is enhanced with in- ¢
creasing frequency. The experimental methods are given in section 2.
Section 3 lists the results and in section 4 the well known RSJ model is used to show that, using cred- ible values for the parameters, one can obtain an ex- cellent qualitative description for nearly all the re- suits. Some of the quantitative problems associated with this model are also outlined.
2. Methods
The techniques used in measuring the temperature and magnetic field dependence of microwave losses are as described before [2,5,6]. In brief, the micron size powders were held on thin (0.16 mm) quartz plates using GE 7031 dilute cement and have fiUing
fractions between 7 and 14 percent. The agglomer- ates were mounted similarly. The thin films were de- posited on LaA1Oa, and again [ 11 ] quartz plates were used to mount them against the cavity wall. The sin- gle crystals were secured in a similar fashion while the pellets were mounted on sapphire rods. In every case, the specimen was placed in a microwave cavity such that bRF was at its maximum value. The T de- pendence of microwave absorption was studied by following the power reflected at the cavity frequency (Pc) as a function of temperature. The zero-f ield loss at any temperature is then defined by P(0, T) =Pc(0, T) - Pc(0, TL). TL marks the temperature below which Pc becomes independent of T. In most cases TL~ 85 K; however, for agglomerates TL~ 65 K. For magnetoabsorption, the sample was cooled to the temperature of interest in zero field (p.oH< 30 IsT). After cooling, the powder and pellet samples were subjected to a slowly ramped DC magnetic field (/zoH) to obtain the virgin curve. Subsequently, the field was slowly cycled between/Zo/-/~.~ and -/zoHm~x (reverse field). Magnetoabsorption in the films and single crystals was studied with bRF in the plane of the specimen and B ( _< 1.5 T) applied perpendicular to it.
Many of the properties of the samples listed in ta- ble 1 have been described in previous reports [2-10] from this laboratory. To summarize, no. 1 was ob- tained from AKZO where they use a proprietary technique, no. 2 was locally prepared using conven- tional heat-grind methods, nos. 3, 4, 5 and 10 were kindly provided by N.D. Spencer from W.R. Grace and Co. and are obtained by co-precipitation from a solution; no. 6, provided by J.E. Crow, consists of 2 × 2 × 1.5 I~m YBCO aligned grains (about 4% by volume) embedded in epoxy, nos. 7 and 8 are 1-2- 3 pellets prepared by Nath et al. [ 14 ] using conven- tional techniques and no. 9 is described in ref. [ 15 ]. The thin films were prepared locally in the Center for Superconductivity Research (for details, see ref. [ 11 ] ) are kindly provided to us by A.T. Findikoglu and T. Venkatesan. The single crystals were fur- nished by J. Lynn (ErBCO, prepared at UC Davis), F. Holzberg (YBCOI, prepared at IBM) and U. Welp (YBCO2, prepared at Argonne National Labs).
J.S. Ramachandran et al. / Microwave absorption
Table 1 Sample characteristics and parameters for eq. ( 1 )
153
Sample Type T c ") Nominal /~/o b) no. (K) diameter (roT)
In addit ion to the field dependences shown in fig. I o f ref. [ 12 ], we display in fig. 1 here the field de- pendence o f magnetoabsorpt ion (on first applica- tion o f field, virgin curve) in a pellet sample (no. 8) at 28.3 K and 10 GHz. The points were calculated using eq. (2) and clearly demonstrate its excellent success. The universal applicability ofeq. (2) can be judged from fig. 2 where the normalized magne- toabsorption A P ( H ) / a is plotted as a function o f
I I I I I I I I
t I I I I f I I 0 0.8 1.6 2.4 3.2 4.0 4.8 5.6
~t0H (roT)
Fig. 1. Field dependence o f the vi rgin magne toabso rp t ion in a pellet specimen of YBCO at l0 GHz and 28.3 K. The solid line represents the data while the open circles represent the fit ob- tained using eq. (2) .
(H/Ho) for all the samples listed in table 1. The val- ues o f rio and c~ obtained from the fits, such as shown in fig. 1, are listed in table I also.
One can obtain a feel for the magnitude o f AP(H) by noting that, at 77 K, application o f 25 mT induces losses which are roughly 10 percent o f the zero-field absorption in the normal state ( T > T¢ + ).
As regards Ho, the following points are notable. (1) Except for samples 1-3 there is no systematic relation between Ho and nominal grain size. (2) Ho is a very weak function o f frequency, the de- crease with increasing frequency (see 10 ~tm data in table 1 ) being barely outside experimental error. (3) Ho reduces monotonically with increasing T, the change being roughly linear between 9 K and To.
As noted previously, fig. 3 which refers to sample no. 4 at 77 K, is typical [ 13 ] o f the hysteretic be- havior observed for all the samples listed in table 1 when the field is ramped linearly between/toHm~, and -/toHm,x. The field separation between the minima Jh and the height o f the min imum absorption Pml, increase with rising Hm,~, and increasing T. However, the detailed functional relationships are still under study and will be reported elsewhere.
Before closing this section it is useful to reiterate that only powders and other obviously grainy sam-
154 J.S. Ramachandran et al. / Microwave absorption
1.0
0.9
0.8
0.7
0.6 tff'(H) - - 0 . 5
0 t
0.4
0.3
0.2
0.1
0 0
IA~f~IS~AI~C~FBI~ABq~¢~ SmF~B K)IB AS(:~IJJo 0 1 I~ U I I
A 123 pellet @ 8.8 K 123 pellet @ 19.1 K
f i 123 pellet @ 69.8 K 123 pellet @ 28.3 K Sample No. I-1 Sample No. I-2 Sample No. I-4
H Sample No. I-5 I Sample No. I-1O J Sample No. I-9 (Ref. 15) a Sample No. I-6
i I i i I i I 5 10 15 20 25 30 35
H/H 0
Fig. 2. The universal applicability ofeq. (2) as shown by plotting the normalized power absorbed as a function of the normalized field in a variety of HTSC samples (table 1 ) on first exposure to the magnetic field.
50 llm YBCO Powder (Sample 1-4) @ ~I0 GHz, 77 K
I 1 [ 1 1 1 --12 -8 --4
P(H) iarb; units)
,l" 0 4 8 12
/.toll (roT)
Fig. 3. Hysteretic magnetoabsorption loop of 50 pan YBCO pow- der by linearly ramping the magnetic field between + 14 mT at 10 GHz and 77 K. The arrows indicate the direction of the sweep of the magnetic field.
pies exhibit the large low-field ( <25-50 mT) mag- netoabsorption and concomitant hysteresis dis- cussed above. High quality films and single crystals are rather insensitive to application of field except at temperatures close to T¢. The high field losses in sin- tie crystals are best ascribed to absorption due to fluxons, as discussed in ref. [ 11 ] for thin films, and the results will be discussed elsewhere.
3.3. T ~ w
In ref. [2 ] it was shown that the onset of super- conductivity is indicated by a sharp drop in Pc pre- ceded by some rounding. Tc was defined as shown in figs. 4(a) , (b) and marks the temperature at which the drastic drop occurs. It was noted in ref. [ 2 ] that
n l w in both cases studied at that time _¢T ma8 > T¢ . In table 2, we have collected all .the data on
T mw along with the corresponding T~ ag values for various samples. The magnetization measurements were made by different groups. T~ m~s for samples II- 1, -2, -3 and -7 was measured by the providers and also confirmed locally. Samples 11-5 and -6 were measured in this laboratory.
Notice that in all cases T~ w < T m~, and that the difference in most cases is larger than the experi- mental error ( + 1 K). This difference is also found to increase with increasing frequency. As an addi- tional check, sample 11-4 was measured using a dif- ferent 10 GHz microwave spectrometer. The con- sequent T mw value agrees with the local measurement to within the experimental error.
3.4. 2'~ w
Reference [3 ] reported on the grain size depen- dence of zero-field microwave absorption in YBCO powders. Using the London model with a simple
J.S. Ramachandran et al. / Microwave absorption 155
Fig. 4 (a) Zero-field microwave power reflected at the cavity frequency as a function of temperature of a layered thin film (sample II-4) at 10 GHz. T~ w (87 K), the microwave "critical" temperature, marks the onset of superconductivity. Note the T ~ (obtained from AC susceptibility) at 91.5 K. (b) Same as in (a), but for a YBCO twinned single crystal (sample II-3 ) at 34 GHz. Again T~ w = 88.6 K while T~ me =92.8 K.
modification, a close fit to the experimental data was secured in all cases. Table 3 shows 2~w obta ined for YBCO powders o f different grain sizes. In all cases it is seen that 2~w is somewhat larger than (2o) mas obtained f rom D C magnet izat ion measurements on the same powders [ 8, I 0 ]. This discrepancy holds for the aligned sample (no. 6) as well.
In ref. [ 11 ] it was shown that using a two-fluid model to fit microwave absorpt ion data in a c-axis
oriented YBCO thin film, yielded (2~w)~=6000_+ 1000 A, much larger than the ac- cepted value o f 1600/k.
4. Discussion
Many authors [ 16 ] have stressed the impor tance o f weak links, inter- or intragranular, to describe the
156 J.S. Ramachandran et al. / Microwave absorption
Table 2 "Critical" temperature values from magnetization and microwave studies
No. Sample Size T~ *~ (K) Microwave frequency (GHz)
T~" (K)
1 YBCO c-axis oriented 3000 A 90.5 10 thin film 36 on LaA103 58
2 ErBCO - 93 10 single crystal 36
3 YBCO2 - 92.8 34 twinned single crystal
4 PrBCO/YBCO/PrBCO/LaAIO3 - 91,5 10 layered thin film 36
5 Bil.ePbo.3Sbo.lSr2Ca2Cu30 lo 6 ixm 110 10 powder 36
6 YBCO powders ~. M2 6 Ixm 92.5 36
f M1 3 Itm 36 7 YBCOI single crystal - 93.5 36
89 86 86 92 89 88.6
87 84
106 102-104 89 89 87-88
Table 3 Penetration depth values from microwave and magnetization studies
No. Sample Diameter (R/~o)mw 2~ w (A) 2 ~ ( A ) (2R, Ixm) (Londonfit)
properties of the high temperature superconducting oxides. Weak links have also been invoked to ac- count for microwave absorption [17 ] and magne- toabsorption [ 18 ]. However, to our knowledge, there is no study in the literature that covers the wide range of phenomena described in sections 1 and 3 and at- tempts to attribute them all to the existence of weak links. As we shall show, a rather simple model is in- deed successful in reproducing the qualitative fea- tures of almost all of the experimental results using quite credible values for the relevant parameters. At the same time, it must be admitted that the com- plexity of the system makes quantitative compari- sons highly elusive.
The starting point is to treat the weak link as a re- sistively shunted Josephson junction (RSJ). Ignor- ing the capacitive term, it is well known [ 19 ] that for an RSJ,
I ~0 d~/ RN 2~ dt
+I¢ sin ¥=Io +Imwe i°~''t (3)
where Ic and RN are the critical current and normal- state resistance of the RSJ, respectively. For the pres- ent application [ 18 ], Io is taken to represent the sur- face current induced by the applied DC magnetic field H and lmw is the microwave current of angular frequency OJmw. In the absence of microwaves the phase ~u would adjust itself such that Ic sin ~Uo=Io.
£S. Ramachandran et aL /Microwave absorption 15 7
When both fields are present one can write ¥=¥o+~=w( t ) and linearize eq. (3) by assuming that ¢'mw (t) ,a~ ¥o. The power loss due to one RSJ can then be written as
1 P=PN 1+1,/2, (4)
where PN= ½I~wR 2 is the normal state, or high field, power absorption in the junction and the parameter r/is given by
~2D~I2 COS 21//0
[~1 1 ~OJmw]2 . (5 ) T ; j
The critical current in an RSJ diminishes as the ex- ternal DC magnetic field is increased and the re- duction factor for an idealized RSJ is the well known diffraction formula [ s in(n~/Oo) ] / (n~/@o) where @o=hc/2e is the flux quantum.
One should treat the sample as a collection of RSJs with a variety of parameters. For instance, the crib ical currents, junction resistances, junction areas and their projections along the applied fields, A.L, can all be expected to have wide variations. For simplicity, it is proposed to regard t/o (the zero field value oft]) as a "lumped" parameters which is common to all the links and write for the total normalized power absorbed as a function of magnetic field
nmax P(x, n) 1 ~ 1
P N - - ( n ~ - - n m i n ) [ nmi~'l 1+17 o2 ( " ) 2 ] d n ' ~ sm nx
(6) where
xP.oH(A.L ) t12= , sin2nx , t18 X= ~o (n.~)
and
I° RN .
tlo =2e (hO)mw) , (7)
while I ° is the zero-field critical current. Since there is no a priori knowledge of the distribution of effec- tive areas, it was decided that one could simulate the specimen by using a fiat distribution. That is, set A.L =n(A.L ), where (A.L) represents an average
value, and integrate over n, as shown in eq. (6). A rough estimate of ~/o can be obtained from the
zero field microwave absorption at the temperature of interest. At 77 K this is typically around 5% of the absorption at T= T~ + except in samples no 1-4 and 1-5 where it was found to be ~ 10%. This implies t?o values between 5 and 3, respectively. For 10 GHz microwaves this gives for the product I°RN (77 K) values of 60--100 gV. This value is much smaller than that expected from the Ambegaokar-Baratoff equa- tion [20 ]
iOcR N rid(T)..A(T) - tann
2e
if~(0) is set equal to I0 meV as is appropriate for YBCO. However, many transport measurements [2 I, 17 ] of the ~RN product in YBCO samples yield results in the range required by the present data. One can therefore proceed to use eqs. (6) and (7) with some confidence.
4.1. Virgin curves
For comparison with experiment one evaluates
_[P(x__n) 1
ot - Pr~-Po - L PN 1 ~2t/~Jk---~o }" (8)
It is to be noted that to get a monotonically varying AP/a, ( n ~ - nmin) > 20 is needed. For smaller val- ues, the function retains vestiges of the oscillatory behavior of the Fraunhofer formula.
Next, the observed values of rio at 77 K were used to estimate the average effective junction areas (A-L). With r/o fixed, eqs. (5) and (8) yield the value o fx (termed)co) for which AP/a = 0.5. This is useful because, empirically (fig. 2), this corresponds to H=Ho. Thus, HoocXo/(A.L). Figure 5 shows xo as a function of t/o, and one notes that for r/o increasing between 1 and 10, )Co rises by a factor of 5. Also in fig. 5 we see that as r/o--,0, )Co--, constant. However, it should be noted that r/o--,0 when 1%-,0 which oc- curs at T~ To. At this temperature (A.L) should be- come extremely large and ensure that Ho--.0.
Using the Ho values given in table 1, it is estimated that at 77 K, (A.L) varies between (0.02-1) gm 2, the smallest values being for the agglomerates and
158 J.S. Ramachandran et al. / Microwave absorption
1.0,
0.8 !
0.6 x 0
0.4
0.2
0 . ( 0 t i i I I i t t i 2 4 6 8 10
q0
Fig. 5. Dependence of the field parameter Xo (where the absorp- tion reaches one-half its saturation value) on ~/o, derived from eqs. (7) and (8). The solid line is a linear fit.
l.O]',,x2--~ t ~ =~ ~ Lll ' ' ~- ~ ' x YBCO Pe et 1 (Sample I-7) !
0.4 ° + ~
0.2 \ " +=+=~=
0 I I I I I I I I I 0.2 0.4 0.6 0.8 1.0
T/W e
Fig. 6. Temperature dependence of the parameter Ho, data shown here are for YBCO pellet 1 (sample I-7 ( l l ) ) and YBCO pellet 2 (sample I-8 ( + ) ) at 10 GHz. The solid line represents Xhe Ho values computed using Ho(T) = H ( 0 ) [ a + b( 1 - t) ] ( 1 - t 4) 1/2 where t=T/Tc and the parameters are a=0.13 and b= l (see text).
the aligned powder (sample no. 1-6). One should ex- pect <A j_ ) ~;td, where d is likely to be of the order of a few ~tm (grain size). Since 2_-__0.3 ~tm, the ob- served <A± > values are quite reasonable.
Experimentally, it is found that Ho decreases monotonically with increasing temperature, as shown in fig. 6. This can be roughly understood from the RSJ model as follows. As we have seen above
Ho ~ :Co <Al > '
with Xo = a + b~/o (I ° ) and <A ± > ~ Ad. The temper-
ature variation of Ho will therefore come from that of 2 and I °. For the former, it is reasonable to use the two-fluid expression of eq. (1). For the latter there are two possibilities. Following Ambegaokar- Baratoff, SIS junctions have I ° ~ ( 1 - t) near Tc while SNS junctions [22] require I ° ~ ( 1 - 0 2. A roughly linear temperature variation for Ho comes about if we use the SIS expression. This is shown schemati- cally by the full line in fig. 6. The parameter values were chosen to conform to the a and b values noted in fig. 5. Thus, present data suggest that the SIS pic- ture is preferable. A similar result was obtained by Kish et al. [23] with regard to a Josephson junction in single crystal YBCO.
From eq. (7) it is seen that r/o should vary in- versely as tOmw. Therefore, the virgin curve, in the frame work of the RSJ model, should be frequency dependent. However, experimentally it is seen that changing tOmw from 10 GHz to 35 GHz (to refer ta- ble 1-10 ~tm YBCO data) gives no significant change in/ tol l 0. The RSJ model is clearly inadequate to ac- count for the observed insensitivity to tO~w.
4.2. Hysteresis
It is useful to rewrite eq. (8) in terms of the field- dependent critical current density J¢(H). That is,
AP 1 - j 2 ( H ) / j 2 ( O ) a - 1 + f l j2 (H) ' (9)
where fl is sample dependent and includes all field independent parameters. From eq. (9) it is expected that ( 1 ) a monotonic increase in magnetoabsorption with increasing H (virgin curve) should correspond to a monotonic decrease in J¢; (2) hysteretic magnetoabsorption should corre- spond to a hysteresis in J~, shown schematically in fig. 7. The characteristics of the hysteresis loop in J~ are expected to be dependent on the maximum field the sample is subjected to (/-/max), such as, a mini- mum in P(H) should correspond to a maximum in Jc(H), and the separation between the maxima in J~(H) peaks should be a function of the extremum fields. Also, the loss in magnetoabsorption on field reduction should correspond to an increase in J~.
Several authors [24] have observed such a hys- teresis in the transport Jc of HTSC materials when
J.S. Ramachandran et al. / Microwave absorption 159
Jc(H)
0
H ~-
Fig. 7. Schematic representation of the critical current density Jc(H). Notice that J , (0)>J~ (0). The arrows represent the di- rections of sweep of the magnetic field (cf. fig. 1, ref. [24] ).
the magnetic field is cycled. It is seen that following the initial (virgin) monotonic reduction as H is in- creased to Hm,~, J~, for reducing H, has a pro- nounced maximum and that J~ (0) is less than J~(0) (see fig. 7). On further field cycling, the J~ versus H graph is seen to be a "hysteresis" loop where the hys- teresis becomes more pronounced if the extremum fields are increased. Although the hysteresis loop in J , (H) is not completely understood from first prin- ciples [25], if the magnetoabsorption follows eq. ( 10 ), Jc (H) will exhibit the kinds of hysteresis loops shown in fig. 7. It is, however, important to note here that while transport measurements require sintered materials, microwaves do not. Therefore, the latter, being perceptive of individual grains, are particu- larly useful in studying powdered HTSC samples.
Thus one can claim that, qualitatively, the hyster- etic magnetoabsorption, of microwaves is grainy samples, is reasonably understood. To obtain quan- titative insights, systematic studies of the Hm~, Tand frequency dependences of the loop parameters Jh and Pm~ are underway at this time.
4.3. TTW < T'~ "as
The magnetoabsorption studies in c-axis thin films and singly crystals at T a few degrees below Tc show very little sensitivity to fields less than 50 mT. Also,
as shown in ref. [ 11 ], the field induced loss is well described by coupling of microwave currents to flux- ons. Thus, at first sight single crystals and thin films appear to be relatively free of weak links. It is, there- fore very surprising that in both thin films and single crystals T~mW # T mag and 8 T c - - ( T m a g - T row) in- creases with increasing microwave frequency (cf. ta- ble 2). The situation is remarkably reminiscent of the behavior [26] of granular Al films.
To understand both of the above observations, it seems reasonable to suggest that very close to To, sin- gle crystals and c-axis oriented thin films of YBCO are, in fact, granular. However, the weak links quickly become inoperative as T drops below To. In YBCO thin films, at least, the percolative nature of the tran- sition has also been demonstrated by Leemann et al. [27]. Kish et al. [23] have also invoked additional weak links in single crystals near T~. It is surmised that the junction energies E ~ k T ~ , so that strong coupling ensues as T drops a few degrees below T~. In other words, the weak links are predominant at T mw < T< Tc mag but become inactive at lower T. In this picture, the weak links first become evident as T drops below T mag so that in the regime T~w< T< T~ "g it is reasonable to write the normalized power absorbed (at zero field) as
P Ps Pwl + PN -- P~, I N '
where Ps represents the (rapidly dropping) absorp- tion in the superconductor, and Pw] is the contri- bution due to the weak links which is, for simplicity, taken to be a J function in T. A non-zero Pw] will de- lay the onset of the sharp drop in the absorption. This ensures that for T< T mw the transition is quite sharp as is, in fact, observed. Also, in the RSJ model (eq. (6) ) Pwl will increase with increasing COmw, thereby rendering a decrease of T mw with increasing fre- quency quite plausible.
In the case of the grainy samples, the weak links appear to be operative at all Tbelow T~, and the non- coincidence of T mw and T~ m~g is not unexpected. The transitions are broader, and this may mask the fre- quency dependence, if any.
4.4. 2'ffw>2~ ae
The sharpness of the drop in microwave absorp- tion at T< Tc is controlled by the effective value of
160 J.S. Ramachandran et al. / Microwave absorption
the London depth 2 and its temperature dependence, eq. ( 1 ). In micron-size powders [ 3 ], using a simple modification of the London's formulae, this was used to obtain a value for (R/;to) row. For the same pow- ders the low-field magnetization was measured [2,8,10], and the temperature dependence of the initial susceptibility was used to calculate (R/Zo) m~. It turned out that ;t~'~ was always somewhat larger than ;t~g. In principle, a more realistic model [28] for the microwave transition yields somewhat smaller A F values. However, the model of ref. [ 28 ] fails to describe the transition as successfully as the simple treatment of ref. [ 3 ]. Be that as it may, the enhanced ;t~w values again point to additional losses due to weak links even in micron-size powders. The most teUing case is that of sample no. 6 where the aligned grains are like parallelepipeds and the magnetic field is along the c-direction so that both the DC and mi- crowave measurements should yield (;to)ab. In real- ity, the former give 2~ "s - 1600 A, as expected, while the microwave results are consistent with ; t F = 2900 A. Again, although the grains are only 2 gm in size they have an anomalously high zero-field absorption pointing to the presence of weak links. The agglom- erates, of course, exhibit very wide transitions and the zero-field absorption at 77 K is largely due to weak links.
5. Summary
Experimental studies of microwave absorption and magnetization studies in CuP based HTSC mate- rials have yielded the following results.
In powders and sintered pellets ( 1 ) the sample's first exposure to low DC magnetic fields (virgin curve) is found to be a universal func- tion described by an extremely simple one-parame- ter equation (eq. (2) ) , (2) the microwave transition width is broader in ag- glomerates as opposed to powders of small grain size (2 ttm to 10 ttm), (3) the characteristics of the hysteresis loop in mag- netoabsorption at low fields is dependent on the maximum field value to which the sample is exposed to and, (4) in powders T~w.. T ~ (5) ;tb"w >;t~ ~ in every case.
In thin films and single crystals (1) there is no measurable magnetoabsorption at field values < 50 roT, (2) T mw is frequency dependent and less than Tmag,
m w m a d (3) in thin films 2o >;to • Most of these results are shown to follow it these
samples are thought to be comprised of weak-links and where each weak-link is modeled as a resistively shunted Josephson junction. Unfortunately, the model falls short in explaining some details such as the frequency independent virgin curve obtained from experiments. However, given the extreme com- plexity of the problem, it is remarkable that a simple RSJ picture is successful in explaining so many of the observed results.
Acknowledgements
We thank N.D. Spencer of W.R. Grace and Co., J.E. Crow of the Center for Materials Research, Flor- ida State University, J. Lynn, T. Venkatesan, A.T. Findikoglu, R.L. Greene and H.D. Drew of the Cen- ter for Superconductivity Research, University of Maryland, for some of the samples. Our thanks are due to A.T. Findikoglu for the T~ m~ measurements of sample no. II-7. We are also grateful to F.C. Wells- tood and M. Manheimer for discussions from which we have greatly benefited.
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